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研究生： 劉懷丞
Huai-Cheng Liu
論文名稱： 數值方法與類神經網路應用於內嵌介面問題
Numerical Methods and Neural Network for Immersed Interface Problem
指導教授： 胡偉帆
Wei-Fan Hu
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 30
中文關鍵詞： 數值分法 、類神經網路 、內嵌介面問題
外文關鍵詞： Numerical Methods, Neural Network, Immersed Interface Problem
相關次數： 點閱：15  下載：0 
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    • 本篇論文會介紹數值分法和類神經網路兩種方式來對偏
    微分方程做數值分析。首先會以中央差分法先來做介紹，而
    這是一個能數值解微分方程的傳統方法。除此之外，也會討
    論針對不同類型的邊界條件所帶來的影響，分別為狄力克雷
    邊界條件和諾伊曼邊界條件。接下來會介紹沈浸介面法，是
    一個能處理任意維度介面問題的方法，但對於高維度問題時
    沈浸介面法會遇到困難。因此，我們會利用類神經網路的方
    法來嘗試能否解決此問題，且得到更好的準確度。針對介面
    問題，將會使用不連續捕獲淺層神經網路法來解決。最後，
    會針對連續與不連續偏的問題做降階法來看是否能得到更好
    的準確度。

    This thesis starts by introducing the Finite Difference Method (FDM)[1]
    which is one of the classic numerical methods that can solve the partial
    differential equations (PDEs). Moreover, we discuss the effect different
    types of boundary conditions on the problems, including Dirichlet and
    Neumann boundary conditions. After that, we will tell the method that
    can solve the d -dimensional interface problem which is, Immersed Interface
    Method (IIM)[2, 3]. Although the IIM could easily solve the interface
    problem, it might be difficult to deal with the high-dimensional problem.
    Therefore, we tried to use the Neural Network (NN) [4] to see whether it
    is a good way to fix it and try to get better accuracy. For the interface
    problem, we used the Discontinuity Capturing Shallow Neural Network
    (DSCNN) [5]. At last, we use a method of reduction for the continuous
    and discontinuous problems to find out it is a way to obtain results better
    than before.

    中文摘要 i 英文摘要 ii 誌謝 iii 目錄 iv 一. Introduction 1 二. Finite Difference Method 2 2.1 One-dimensional Boundary Value Problem 2 2.2 Immersed Interface Method for One-dimensional Interface Problem 4 三. Neural Network 6 3.1 Neural Network structure 6 3.2 Discontinuity Capturing Shallow Neural Network structure 7 3.3 Function approximation 9 3.4 Solving Poisson equation 9 3.5 Solving Elliptic Interface Problems 12 3.6 Higher order problems 17 四. Conclusion 20 五. References 20

    [1] R. J. LeVeque, Finite Difference Methods for Differential Equations, SIAM, (2005).
    [2] Z. Li, K. Ito, The Immersed Interface Method: Numerical Solutions of PDEs InvolvingInterfaces and Irregular Domains, SIAM, (2006).
    [3] H.-C. Tseng, Numerical Methods and Applications for Immersed Interface Problems, Na-tional Chiao Tung University, (2006).
    [4] C. F. Higham, D. J. Higham, Deep Learning: An Introduction for Applied Mathematics,SIAM, (2019).
    [5] W.-F. Hu,T.-S. Lin ,M.-C. Lai, A Discontinuity Capturing shallow Neural Network for Elliptic Interface Problems, arXiv : 2106.05587, (2021).
    [6] D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11(2) (431-441).
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